begin
theorem
for
n being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
for
A,
B being ( ( non
empty compact ) ( non
empty functional closed compact )
Subset of )
for
f being ( (
Function-like V35( the
carrier of
[:(TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) ,(TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) :] : ( (
strict TopSpace-like ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) )
continuous ) (
V21()
V24( the
carrier of
[:(TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) ,(TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) :] : ( (
strict TopSpace-like ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set ) )
V25(
REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) )
Function-like V35( the
carrier of
[:(TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) ,(TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) :] : ( (
strict TopSpace-like ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) )
continuous )
RealMap of ( ( ) ( non
empty )
set ) )
for
g being ( (
Function-like V35( the
carrier of
(TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) ,
REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) ) ) (
V21()
V24( the
carrier of
(TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) )
V25(
REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) )
Function-like V35( the
carrier of
(TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) ,
REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) ) )
RealMap of ( ( ) ( non
empty functional )
set ) ) st ( for
p being ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(
b1 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Point of ( ( ) ( non
empty functional )
set ) ) ex
G being ( ( ) (
V147()
V148()
V149() )
Subset of ( ( ) ( non
empty )
set ) ) st
(
G : ( ( ) (
V147()
V148()
V149() )
Subset of ( ( ) ( non
empty )
set ) )
= { (f : ( ( Function-like V35( the carrier of [:(TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) ,(TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) ) continuous ) ( V21() V24( the carrier of [:(TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) ,(TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) ) V25( REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) ) Function-like V35( the carrier of [:(TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) ,(TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) ) continuous ) RealMap of ( ( ) ( non empty ) set ) ) . (p : ( ( ) ( V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) Function-like finite V45(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V139() ) Point of ( ( ) ( non empty functional ) set ) ) ,q : ( ( ) ( V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) Function-like finite V45(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V139() ) Point of ( ( ) ( non empty functional ) set ) ) )) : ( ( ) ( ) set ) where q is ( ( ) ( V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) Function-like finite V45(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V139() ) Point of ( ( ) ( non empty functional ) set ) ) : q : ( ( ) ( V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) Function-like finite V45(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V139() ) Point of ( ( ) ( non empty functional ) set ) ) in B : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) } &
g : ( (
Function-like V35( the
carrier of
(TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) ,
REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) ) ) (
V21()
V24( the
carrier of
(TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) )
V25(
REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) )
Function-like V35( the
carrier of
(TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) ,
REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) ) )
RealMap of ( ( ) ( non
empty functional )
set ) )
. p : ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(
b1 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Point of ( ( ) ( non
empty functional )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) )
= lower_bound G : ( ( ) (
V147()
V148()
V149() )
Subset of ( ( ) ( non
empty )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) ) ) ) holds
lower_bound (f : ( ( Function-like V35( the carrier of [:(TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) ,(TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) ) continuous ) ( V21() V24( the carrier of [:(TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) ,(TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) ) V25( REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) ) Function-like V35( the carrier of [:(TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) ,(TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) ) continuous ) RealMap of ( ( ) ( non empty ) set ) ) .: [:A : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) ,B : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) :] : ( ( ) ( non empty ) Element of bool the carrier of [:(TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) ,(TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non
empty V147()
V148()
V149() )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) )
= lower_bound (g : ( ( Function-like V35( the carrier of (TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) : ( ( ) ( non empty functional ) set ) , REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) ) ) ( V21() V24( the carrier of (TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) : ( ( ) ( non empty functional ) set ) ) V25( REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) ) Function-like V35( the carrier of (TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) : ( ( ) ( non empty functional ) set ) , REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) ) ) RealMap of ( ( ) ( non empty functional ) set ) ) .: A : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) ) : ( ( ) ( non
empty V147()
V148()
V149() )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) ) ;
theorem
for
n being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
for
A,
B being ( ( non
empty compact ) ( non
empty functional closed compact )
Subset of )
for
f being ( (
Function-like V35( the
carrier of
[:(TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) ,(TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) :] : ( (
strict TopSpace-like ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) )
continuous ) (
V21()
V24( the
carrier of
[:(TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) ,(TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) :] : ( (
strict TopSpace-like ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set ) )
V25(
REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) )
Function-like V35( the
carrier of
[:(TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) ,(TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) :] : ( (
strict TopSpace-like ) ( non
empty strict TopSpace-like )
TopStruct ) : ( ( ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) )
continuous )
RealMap of ( ( ) ( non
empty )
set ) )
for
g being ( (
Function-like V35( the
carrier of
(TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) ,
REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) ) ) (
V21()
V24( the
carrier of
(TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) )
V25(
REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) )
Function-like V35( the
carrier of
(TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) ,
REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) ) )
RealMap of ( ( ) ( non
empty functional )
set ) ) st ( for
q being ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(
b1 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Point of ( ( ) ( non
empty functional )
set ) ) ex
G being ( ( ) (
V147()
V148()
V149() )
Subset of ( ( ) ( non
empty )
set ) ) st
(
G : ( ( ) (
V147()
V148()
V149() )
Subset of ( ( ) ( non
empty )
set ) )
= { (f : ( ( Function-like V35( the carrier of [:(TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) ,(TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) ) continuous ) ( V21() V24( the carrier of [:(TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) ,(TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) ) V25( REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) ) Function-like V35( the carrier of [:(TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) ,(TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) ) continuous ) RealMap of ( ( ) ( non empty ) set ) ) . (p : ( ( ) ( V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) Function-like finite V45(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V139() ) Point of ( ( ) ( non empty functional ) set ) ) ,q : ( ( ) ( V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) Function-like finite V45(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V139() ) Point of ( ( ) ( non empty functional ) set ) ) )) : ( ( ) ( ) set ) where p is ( ( ) ( V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) Function-like finite V45(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V139() ) Point of ( ( ) ( non empty functional ) set ) ) : p : ( ( ) ( V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) Function-like finite V45(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V139() ) Point of ( ( ) ( non empty functional ) set ) ) in A : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) } &
g : ( (
Function-like V35( the
carrier of
(TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) ,
REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) ) ) (
V21()
V24( the
carrier of
(TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) )
V25(
REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) )
Function-like V35( the
carrier of
(TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) ,
REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) ) )
RealMap of ( ( ) ( non
empty functional )
set ) )
. q : ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(
b1 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Point of ( ( ) ( non
empty functional )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) )
= lower_bound G : ( ( ) (
V147()
V148()
V149() )
Subset of ( ( ) ( non
empty )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) ) ) ) holds
lower_bound (f : ( ( Function-like V35( the carrier of [:(TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) ,(TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) ) continuous ) ( V21() V24( the carrier of [:(TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) ,(TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) ) V25( REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) ) Function-like V35( the carrier of [:(TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) ,(TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) ) continuous ) RealMap of ( ( ) ( non empty ) set ) ) .: [:A : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) ,B : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) :] : ( ( ) ( non empty ) Element of bool the carrier of [:(TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) ,(TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non
empty V147()
V148()
V149() )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) )
= lower_bound (g : ( ( Function-like V35( the carrier of (TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) : ( ( ) ( non empty functional ) set ) , REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) ) ) ( V21() V24( the carrier of (TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) : ( ( ) ( non empty functional ) set ) ) V25( REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) ) Function-like V35( the carrier of (TOP-REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) : ( ( ) ( non empty functional ) set ) , REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) ) ) RealMap of ( ( ) ( non empty functional ) set ) ) .: B : ( ( non empty compact ) ( non empty functional closed compact ) Subset of ) ) : ( ( ) ( non
empty V147()
V148()
V149() )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) ) ;
begin
definition
let n be ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) ;
func Eucl_dist n -> ( (
Function-like V35( the
carrier of
[:(TOP-REAL n : ( ( ) ( ) MetrStruct ) ) : ( ( V171() ) ( V171() ) L15()) ,(TOP-REAL n : ( ( ) ( ) MetrStruct ) ) : ( ( V171() ) ( V171() ) L15()) :] : ( (
strict TopSpace-like ) (
strict TopSpace-like )
TopStruct ) : ( ( ) ( )
set ) ,
REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) ) ) (
V21()
V24( the
carrier of
[:(TOP-REAL n : ( ( ) ( ) MetrStruct ) ) : ( ( V171() ) ( V171() ) L15()) ,(TOP-REAL n : ( ( ) ( ) MetrStruct ) ) : ( ( V171() ) ( V171() ) L15()) :] : ( (
strict TopSpace-like ) (
strict TopSpace-like )
TopStruct ) : ( ( ) ( )
set ) )
V25(
REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) )
Function-like V35( the
carrier of
[:(TOP-REAL n : ( ( ) ( ) MetrStruct ) ) : ( ( V171() ) ( V171() ) L15()) ,(TOP-REAL n : ( ( ) ( ) MetrStruct ) ) : ( ( V171() ) ( V171() ) L15()) :] : ( (
strict TopSpace-like ) (
strict TopSpace-like )
TopStruct ) : ( ( ) ( )
set ) ,
REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) ) )
RealMap of ( ( ) ( )
set ) )
means
for
p,
q being ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(
n : ( ( ) ( )
MetrStruct ) )
FinSequence-like FinSubsequence-like V139() )
Point of ( ( ) ( )
set ) ) holds
it : ( (
Function-like V35(
[:n : ( ( ) ( ) MetrStruct ) ,n : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( )
set ) ,
REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) ) ) (
V21()
V24(
[:n : ( ( ) ( ) MetrStruct ) ,n : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( )
set ) )
V25(
REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) )
Function-like V35(
[:n : ( ( ) ( ) MetrStruct ) ,n : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( )
set ) ,
REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) ) )
Element of
bool [:[:n : ( ( ) ( ) MetrStruct ) ,n : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) :] : ( ( ) ( )
set ) : ( ( ) ( non
empty )
set ) )
. (
p : ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(
n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Point of ( ( ) ( non
empty functional )
set ) ) ,
q : ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(
n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Point of ( ( ) ( non
empty functional )
set ) ) ) : ( ( ) ( )
set )
= |.(p : ( ( ) ( V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) Function-like finite V45(n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V139() ) Point of ( ( ) ( non empty functional ) set ) ) - q : ( ( ) ( V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) Function-like finite V45(n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V139() ) Point of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( ) Element of the carrier of (TOP-REAL n : ( ( ) ( ) MetrStruct ) ) : ( ( V171() ) ( V171() ) L15()) : ( ( ) ( ) set ) ) .| : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) ) ;
end;
definition
let T be ( ( non
empty TopSpace-like ) ( non
empty TopSpace-like )
TopSpace) ;
let f be ( (
Function-like V35( the
carrier of
T : ( ( non
empty TopSpace-like ) ( non
empty TopSpace-like )
TopSpace) : ( ( ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) ) ) (
V21()
V24( the
carrier of
T : ( ( non
empty TopSpace-like ) ( non
empty TopSpace-like )
TopSpace) : ( ( ) ( non
empty )
set ) )
V25(
REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) )
Function-like V35( the
carrier of
T : ( ( non
empty TopSpace-like ) ( non
empty TopSpace-like )
TopSpace) : ( ( ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) ) )
RealMap of ( ( ) ( non
empty )
set ) ) ;
redefine attr f is
continuous means
for
p being ( ( ) ( )
Point of ( ( ) ( )
set ) )
for
N being ( ( ) (
open V147()
V148()
V149() )
Neighbourhood of
f : ( (
Function-like V35(
[:T : ( ( ) ( ) MetrStruct ) ,T : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( )
set ) ,
REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) ) ) (
V21()
V24(
[:T : ( ( ) ( ) MetrStruct ) ,T : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( )
set ) )
V25(
REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) )
Function-like V35(
[:T : ( ( ) ( ) MetrStruct ) ,T : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( )
set ) ,
REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) ) )
Element of
bool [:[:T : ( ( ) ( ) MetrStruct ) ,T : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) :] : ( ( ) ( )
set ) : ( ( ) ( non
empty )
set ) )
. p : ( ( ) ( )
Point of ( ( ) ( non
empty )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) ) ) ex
V being ( ( ) ( )
a_neighborhood of
p : ( ( ) ( )
Point of ( ( ) ( non
empty )
set ) ) ) st
f : ( (
Function-like V35(
[:T : ( ( ) ( ) MetrStruct ) ,T : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( )
set ) ,
REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) ) ) (
V21()
V24(
[:T : ( ( ) ( ) MetrStruct ) ,T : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( )
set ) )
V25(
REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) )
Function-like V35(
[:T : ( ( ) ( ) MetrStruct ) ,T : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( )
set ) ,
REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) ) )
Element of
bool [:[:T : ( ( ) ( ) MetrStruct ) ,T : ( ( ) ( ) MetrStruct ) :] : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) :] : ( ( ) ( )
set ) : ( ( ) ( non
empty )
set ) )
.: V : ( ( ) ( )
a_neighborhood of
b1 : ( ( ) ( )
Point of ( ( ) ( non
empty )
set ) ) ) : ( ( ) (
V147()
V148()
V149() )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) )
c= N : ( ( ) (
open V147()
V148()
V149() )
Neighbourhood of
f : ( (
Function-like V35( the
carrier of
T : ( ( non
empty TopSpace-like ) ( non
empty TopSpace-like )
TopSpace) : ( ( ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) ) ) (
V21()
V24( the
carrier of
T : ( ( non
empty TopSpace-like ) ( non
empty TopSpace-like )
TopSpace) : ( ( ) ( non
empty )
set ) )
V25(
REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) )
Function-like V35( the
carrier of
T : ( ( non
empty TopSpace-like ) ( non
empty TopSpace-like )
TopSpace) : ( ( ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) ) )
RealMap of ( ( ) ( non
empty )
set ) )
. b1 : ( ( ) ( )
Point of ( ( ) ( non
empty )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) ) ) ;
end;
begin
begin
theorem
for
C being ( (
being_simple_closed_curve ) ( non
empty functional closed being_simple_closed_curve compact )
Simple_closed_curve)
for
p,
q being ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(2 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Point of ( ( ) ( non
empty functional )
set ) ) st
LE p : ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(2 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Point of ( ( ) ( non
empty functional )
set ) ) ,
q : ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(2 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Point of ( ( ) ( non
empty functional )
set ) ) ,
C : ( (
being_simple_closed_curve ) ( non
empty functional closed being_simple_closed_curve compact )
Simple_closed_curve) &
LE q : ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(2 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Point of ( ( ) ( non
empty functional )
set ) ) ,
E-max C : ( (
being_simple_closed_curve ) ( non
empty functional closed being_simple_closed_curve compact )
Simple_closed_curve) : ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(2 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Element of the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) ) ,
C : ( (
being_simple_closed_curve ) ( non
empty functional closed being_simple_closed_curve compact )
Simple_closed_curve) &
p : ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(2 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Point of ( ( ) ( non
empty functional )
set ) )
<> q : ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(2 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Point of ( ( ) ( non
empty functional )
set ) ) holds
Segment (
p : ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(2 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Point of ( ( ) ( non
empty functional )
set ) ) ,
q : ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(2 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Point of ( ( ) ( non
empty functional )
set ) ) ,
C : ( (
being_simple_closed_curve ) ( non
empty functional closed being_simple_closed_curve compact )
Simple_closed_curve) ) : ( ( ) (
functional )
Element of
bool the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) : ( ( ) ( non
empty )
set ) )
= Segment (
(Upper_Arc C : ( ( being_simple_closed_curve ) ( non empty functional closed being_simple_closed_curve compact ) Simple_closed_curve) ) : ( ( non
empty ) ( non
empty functional )
Element of
bool the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) : ( ( ) ( non
empty )
set ) ) ,
(W-min C : ( ( being_simple_closed_curve ) ( non empty functional closed being_simple_closed_curve compact ) Simple_closed_curve) ) : ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(2 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Element of the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) ) ,
(E-max C : ( ( being_simple_closed_curve ) ( non empty functional closed being_simple_closed_curve compact ) Simple_closed_curve) ) : ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(2 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Element of the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) ) ,
p : ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(2 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Point of ( ( ) ( non
empty functional )
set ) ) ,
q : ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(2 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Point of ( ( ) ( non
empty functional )
set ) ) ) : ( ( ) (
functional )
Element of
bool the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) : ( ( ) ( non
empty )
set ) ) ;
theorem
for
C being ( (
being_simple_closed_curve ) ( non
empty functional closed being_simple_closed_curve compact )
Simple_closed_curve)
for
p,
q being ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(2 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Point of ( ( ) ( non
empty functional )
set ) ) st
LE p : ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(2 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Point of ( ( ) ( non
empty functional )
set ) ) ,
q : ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(2 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Point of ( ( ) ( non
empty functional )
set ) ) ,
C : ( (
being_simple_closed_curve ) ( non
empty functional closed being_simple_closed_curve compact )
Simple_closed_curve) &
LE E-max C : ( (
being_simple_closed_curve ) ( non
empty functional closed being_simple_closed_curve compact )
Simple_closed_curve) : ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(2 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Element of the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) ) ,
p : ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(2 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Point of ( ( ) ( non
empty functional )
set ) ) ,
C : ( (
being_simple_closed_curve ) ( non
empty functional closed being_simple_closed_curve compact )
Simple_closed_curve) holds
Segment (
p : ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(2 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Point of ( ( ) ( non
empty functional )
set ) ) ,
q : ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(2 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Point of ( ( ) ( non
empty functional )
set ) ) ,
C : ( (
being_simple_closed_curve ) ( non
empty functional closed being_simple_closed_curve compact )
Simple_closed_curve) ) : ( ( ) (
functional )
Element of
bool the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) : ( ( ) ( non
empty )
set ) )
= Segment (
(Lower_Arc C : ( ( being_simple_closed_curve ) ( non empty functional closed being_simple_closed_curve compact ) Simple_closed_curve) ) : ( ( non
empty ) ( non
empty functional )
Element of
bool the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) : ( ( ) ( non
empty )
set ) ) ,
(E-max C : ( ( being_simple_closed_curve ) ( non empty functional closed being_simple_closed_curve compact ) Simple_closed_curve) ) : ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(2 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Element of the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) ) ,
(W-min C : ( ( being_simple_closed_curve ) ( non empty functional closed being_simple_closed_curve compact ) Simple_closed_curve) ) : ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(2 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Element of the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) ) ,
p : ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(2 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Point of ( ( ) ( non
empty functional )
set ) ) ,
q : ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(2 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Point of ( ( ) ( non
empty functional )
set ) ) ) : ( ( ) (
functional )
Element of
bool the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) : ( ( ) ( non
empty )
set ) ) ;
theorem
for
C being ( (
being_simple_closed_curve ) ( non
empty functional closed being_simple_closed_curve compact )
Simple_closed_curve)
for
p,
q being ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(2 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Point of ( ( ) ( non
empty functional )
set ) ) st
LE p : ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(2 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Point of ( ( ) ( non
empty functional )
set ) ) ,
E-max C : ( (
being_simple_closed_curve ) ( non
empty functional closed being_simple_closed_curve compact )
Simple_closed_curve) : ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(2 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Element of the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) ) ,
C : ( (
being_simple_closed_curve ) ( non
empty functional closed being_simple_closed_curve compact )
Simple_closed_curve) &
LE E-max C : ( (
being_simple_closed_curve ) ( non
empty functional closed being_simple_closed_curve compact )
Simple_closed_curve) : ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(2 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Element of the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) ) ,
q : ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(2 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Point of ( ( ) ( non
empty functional )
set ) ) ,
C : ( (
being_simple_closed_curve ) ( non
empty functional closed being_simple_closed_curve compact )
Simple_closed_curve) holds
Segment (
p : ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(2 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Point of ( ( ) ( non
empty functional )
set ) ) ,
q : ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(2 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Point of ( ( ) ( non
empty functional )
set ) ) ,
C : ( (
being_simple_closed_curve ) ( non
empty functional closed being_simple_closed_curve compact )
Simple_closed_curve) ) : ( ( ) (
functional )
Element of
bool the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) : ( ( ) ( non
empty )
set ) )
= (R_Segment ((Upper_Arc C : ( ( being_simple_closed_curve ) ( non empty functional closed being_simple_closed_curve compact ) Simple_closed_curve) ) : ( ( non empty ) ( non empty functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) : ( ( ) ( non empty functional ) set ) : ( ( ) ( non empty ) set ) ) ,(W-min C : ( ( being_simple_closed_curve ) ( non empty functional closed being_simple_closed_curve compact ) Simple_closed_curve) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) Function-like finite V45(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V139() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,(E-max C : ( ( being_simple_closed_curve ) ( non empty functional closed being_simple_closed_curve compact ) Simple_closed_curve) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) Function-like finite V45(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V139() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,p : ( ( ) ( V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) Function-like finite V45(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V139() ) Point of ( ( ) ( non empty functional ) set ) ) )) : ( ( ) (
functional )
Element of
bool the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) : ( ( ) ( non
empty )
set ) )
\/ (L_Segment ((Lower_Arc C : ( ( being_simple_closed_curve ) ( non empty functional closed being_simple_closed_curve compact ) Simple_closed_curve) ) : ( ( non empty ) ( non empty functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) : ( ( ) ( non empty functional ) set ) : ( ( ) ( non empty ) set ) ) ,(E-max C : ( ( being_simple_closed_curve ) ( non empty functional closed being_simple_closed_curve compact ) Simple_closed_curve) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) Function-like finite V45(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V139() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,(W-min C : ( ( being_simple_closed_curve ) ( non empty functional closed being_simple_closed_curve compact ) Simple_closed_curve) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) Function-like finite V45(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V139() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,q : ( ( ) ( V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) Function-like finite V45(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V139() ) Point of ( ( ) ( non empty functional ) set ) ) )) : ( ( ) (
functional )
Element of
bool the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) : ( ( ) ( non
empty )
set ) ) : ( ( ) (
functional )
Element of
bool the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) : ( ( ) ( non
empty )
set ) ) ;
theorem
for
C being ( (
being_simple_closed_curve ) ( non
empty functional closed being_simple_closed_curve compact )
Simple_closed_curve)
for
p being ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(2 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Point of ( ( ) ( non
empty functional )
set ) ) st
LE p : ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(2 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Point of ( ( ) ( non
empty functional )
set ) ) ,
E-max C : ( (
being_simple_closed_curve ) ( non
empty functional closed being_simple_closed_curve compact )
Simple_closed_curve) : ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(2 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Element of the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) ) ,
C : ( (
being_simple_closed_curve ) ( non
empty functional closed being_simple_closed_curve compact )
Simple_closed_curve) holds
Segment (
p : ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(2 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Point of ( ( ) ( non
empty functional )
set ) ) ,
(W-min C : ( ( being_simple_closed_curve ) ( non empty functional closed being_simple_closed_curve compact ) Simple_closed_curve) ) : ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(2 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Element of the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) ) ,
C : ( (
being_simple_closed_curve ) ( non
empty functional closed being_simple_closed_curve compact )
Simple_closed_curve) ) : ( ( ) (
functional )
Element of
bool the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) : ( ( ) ( non
empty )
set ) )
= (R_Segment ((Upper_Arc C : ( ( being_simple_closed_curve ) ( non empty functional closed being_simple_closed_curve compact ) Simple_closed_curve) ) : ( ( non empty ) ( non empty functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) : ( ( ) ( non empty functional ) set ) : ( ( ) ( non empty ) set ) ) ,(W-min C : ( ( being_simple_closed_curve ) ( non empty functional closed being_simple_closed_curve compact ) Simple_closed_curve) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) Function-like finite V45(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V139() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,(E-max C : ( ( being_simple_closed_curve ) ( non empty functional closed being_simple_closed_curve compact ) Simple_closed_curve) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) Function-like finite V45(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V139() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,p : ( ( ) ( V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) Function-like finite V45(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V139() ) Point of ( ( ) ( non empty functional ) set ) ) )) : ( ( ) (
functional )
Element of
bool the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) : ( ( ) ( non
empty )
set ) )
\/ (L_Segment ((Lower_Arc C : ( ( being_simple_closed_curve ) ( non empty functional closed being_simple_closed_curve compact ) Simple_closed_curve) ) : ( ( non empty ) ( non empty functional ) Element of bool the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) : ( ( ) ( non empty functional ) set ) : ( ( ) ( non empty ) set ) ) ,(E-max C : ( ( being_simple_closed_curve ) ( non empty functional closed being_simple_closed_curve compact ) Simple_closed_curve) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) Function-like finite V45(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V139() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,(W-min C : ( ( being_simple_closed_curve ) ( non empty functional closed being_simple_closed_curve compact ) Simple_closed_curve) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) Function-like finite V45(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V139() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,(W-min C : ( ( being_simple_closed_curve ) ( non empty functional closed being_simple_closed_curve compact ) Simple_closed_curve) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) Function-like finite V45(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V139() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) : ( ( ) ( non empty functional ) set ) ) )) : ( ( ) (
functional )
Element of
bool the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) : ( ( ) ( non
empty )
set ) ) : ( ( ) (
functional )
Element of
bool the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) : ( ( ) ( non
empty )
set ) ) ;
begin
definition
let C be ( (
being_simple_closed_curve ) ( non
empty functional closed being_simple_closed_curve compact )
Simple_closed_curve) ;
mode Segmentation of
C -> ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
V25( the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) )
Function-like finite FinSequence-like FinSubsequence-like )
FinSequence of ( ( ) ( non
empty functional )
set ) )
means
(
it : ( (
V21()
Function-like ) (
V21()
Function-like )
set )
/. 1 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) : ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(2 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Element of the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) )
= W-min C : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) : ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(2 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Element of the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) ) &
it : ( (
V21()
Function-like ) (
V21()
Function-like )
set ) is
one-to-one & 8 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
<= len it : ( (
V21()
Function-like ) (
V21()
Function-like )
set ) : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) &
rng it : ( (
V21()
Function-like ) (
V21()
Function-like )
set ) : ( ( ) (
functional )
Element of
bool the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) : ( ( ) ( non
empty )
set ) )
c= C : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) & ( for
i being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) st 1 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
<= i : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) &
i : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
< len it : ( (
V21()
Function-like ) (
V21()
Function-like )
set ) : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) holds
LE it : ( (
V21()
Function-like ) (
V21()
Function-like )
set )
/. i : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) : ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(2 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Element of the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) ) ,
it : ( (
V21()
Function-like ) (
V21()
Function-like )
set )
/. (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) : ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(2 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Element of the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) ) ,
C : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) ) & ( for
i being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) st 1 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
<= i : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) &
i : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
+ 1 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
< len it : ( (
V21()
Function-like ) (
V21()
Function-like )
set ) : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) holds
(Segment ((it : ( ( V21() Function-like ) ( V21() Function-like ) set ) /. i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) Function-like finite V45(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V139() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,(it : ( ( V21() Function-like ) ( V21() Function-like ) set ) /. (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) Function-like finite V45(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V139() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,C : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( ) (
functional )
Element of
bool the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) : ( ( ) ( non
empty )
set ) )
/\ (Segment ((it : ( ( V21() Function-like ) ( V21() Function-like ) set ) /. (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) Function-like finite V45(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V139() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,(it : ( ( V21() Function-like ) ( V21() Function-like ) set ) /. (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) + 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) Function-like finite V45(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V139() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,C : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( ) (
functional )
Element of
bool the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) : ( ( ) ( non
empty )
set ) ) : ( ( ) (
functional )
Element of
bool the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) : ( ( ) ( non
empty )
set ) )
= {(it : ( ( V21() Function-like ) ( V21() Function-like ) set ) /. (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) Function-like finite V45(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V139() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) : ( ( ) ( non empty functional ) set ) ) } : ( ( ) ( non
empty trivial functional finite V42() )
set ) ) &
(Segment ((it : ( ( V21() Function-like ) ( V21() Function-like ) set ) /. (len it : ( ( V21() Function-like ) ( V21() Function-like ) set ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) Function-like finite V45(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V139() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,(it : ( ( V21() Function-like ) ( V21() Function-like ) set ) /. 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) Function-like finite V45(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V139() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,C : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( ) (
functional )
Element of
bool the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) : ( ( ) ( non
empty )
set ) )
/\ (Segment ((it : ( ( V21() Function-like ) ( V21() Function-like ) set ) /. 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) Function-like finite V45(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V139() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,(it : ( ( V21() Function-like ) ( V21() Function-like ) set ) /. 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) Function-like finite V45(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V139() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,C : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( ) (
functional )
Element of
bool the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) : ( ( ) ( non
empty )
set ) ) : ( ( ) (
functional )
Element of
bool the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) : ( ( ) ( non
empty )
set ) )
= {(it : ( ( V21() Function-like ) ( V21() Function-like ) set ) /. 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) Function-like finite V45(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V139() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) : ( ( ) ( non empty functional ) set ) ) } : ( ( ) ( non
empty trivial functional finite V42() )
set ) &
(Segment ((it : ( ( V21() Function-like ) ( V21() Function-like ) set ) /. ((len it : ( ( V21() Function-like ) ( V21() Function-like ) set ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) -' 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) Function-like finite V45(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V139() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,(it : ( ( V21() Function-like ) ( V21() Function-like ) set ) /. (len it : ( ( V21() Function-like ) ( V21() Function-like ) set ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) Function-like finite V45(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V139() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,C : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( ) (
functional )
Element of
bool the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) : ( ( ) ( non
empty )
set ) )
/\ (Segment ((it : ( ( V21() Function-like ) ( V21() Function-like ) set ) /. (len it : ( ( V21() Function-like ) ( V21() Function-like ) set ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) Function-like finite V45(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V139() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,(it : ( ( V21() Function-like ) ( V21() Function-like ) set ) /. 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) Function-like finite V45(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V139() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) : ( ( ) ( non empty functional ) set ) ) ,C : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( ) (
functional )
Element of
bool the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) : ( ( ) ( non
empty )
set ) ) : ( ( ) (
functional )
Element of
bool the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) : ( ( ) ( non
empty )
set ) )
= {(it : ( ( V21() Function-like ) ( V21() Function-like ) set ) /. (len it : ( ( V21() Function-like ) ( V21() Function-like ) set ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V21() V24( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) Function-like finite V45(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V139() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( V171() ) ( non empty TopSpace-like T_0 T_1 T_2 V113() V159() V160() V161() V162() V163() V164() V165() V171() ) L15()) : ( ( ) ( non empty functional ) set ) ) } : ( ( ) ( non
empty trivial functional finite V42() )
set ) &
Segment (
(it : ( ( V21() Function-like ) ( V21() Function-like ) set ) /. ((len it : ( ( V21() Function-like ) ( V21() Function-like ) set ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) -' 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(2 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Element of the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) ) ,
(it : ( ( V21() Function-like ) ( V21() Function-like ) set ) /. (len it : ( ( V21() Function-like ) ( V21() Function-like ) set ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(2 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Element of the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) ) ,
C : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) ) : ( ( ) (
functional )
Element of
bool the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) : ( ( ) ( non
empty )
set ) )
misses Segment (
(it : ( ( V21() Function-like ) ( V21() Function-like ) set ) /. 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(2 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Element of the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) ) ,
(it : ( ( V21() Function-like ) ( V21() Function-like ) set ) /. 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(2 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Element of the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) ) ,
C : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) ) : ( ( ) (
functional )
Element of
bool the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) : ( ( ) ( non
empty )
set ) ) & ( for
i,
j being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) st 1 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
<= i : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) &
i : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
< j : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) &
j : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
< len it : ( (
V21()
Function-like ) (
V21()
Function-like )
set ) : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) & not
i : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) ,
j : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
are_adjacent1 holds
Segment (
(it : ( ( V21() Function-like ) ( V21() Function-like ) set ) /. i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(2 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Element of the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) ) ,
(it : ( ( V21() Function-like ) ( V21() Function-like ) set ) /. (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(2 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Element of the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) ) ,
C : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) ) : ( ( ) (
functional )
Element of
bool the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) : ( ( ) ( non
empty )
set ) )
misses Segment (
(it : ( ( V21() Function-like ) ( V21() Function-like ) set ) /. j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(2 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Element of the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) ) ,
(it : ( ( V21() Function-like ) ( V21() Function-like ) set ) /. (j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(2 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Element of the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) ) ,
C : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) ) : ( ( ) (
functional )
Element of
bool the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) : ( ( ) ( non
empty )
set ) ) ) & ( for
i being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) st 1 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
< i : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) &
i : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
+ 1 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
< len it : ( (
V21()
Function-like ) (
V21()
Function-like )
set ) : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) holds
Segment (
(it : ( ( V21() Function-like ) ( V21() Function-like ) set ) /. (len it : ( ( V21() Function-like ) ( V21() Function-like ) set ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(2 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Element of the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) ) ,
(it : ( ( V21() Function-like ) ( V21() Function-like ) set ) /. 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(2 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Element of the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) ) ,
C : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) ) : ( ( ) (
functional )
Element of
bool the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) : ( ( ) ( non
empty )
set ) )
misses Segment (
(it : ( ( V21() Function-like ) ( V21() Function-like ) set ) /. i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(2 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Element of the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) ) ,
(it : ( ( V21() Function-like ) ( V21() Function-like ) set ) /. (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) (
V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
Function-like finite V45(2 : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
left_end bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) )
FinSequence-like FinSubsequence-like V139() )
Element of the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) ) ,
C : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) ) ) : ( ( ) (
functional )
Element of
bool the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) : ( ( ) ( non
empty )
set ) ) ) );
end;
begin
begin
begin
definition
let C be ( (
being_simple_closed_curve ) ( non
empty functional closed being_simple_closed_curve compact )
Simple_closed_curve) ;
let S be ( ( ) ( non
empty non
trivial V21()
V24(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
V25( the
carrier of
(TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( (
V171() ) ( non
empty TopSpace-like T_0 T_1 T_2 V113()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V171() )
L15()) : ( ( ) ( non
empty functional )
set ) )
Function-like finite FinSequence-like FinSubsequence-like )
Segmentation of
C : ( (
being_simple_closed_curve ) ( non
empty functional closed being_simple_closed_curve compact )
Simple_closed_curve) ) ;
func S-Gap S -> ( ( ) (
V11()
real ext-real )
Real)
means
ex
S1,
S2 being ( ( non
empty finite ) ( non
empty compact closed finite V147()
V148()
V149()
left_end right_end bounded_below bounded_above real-bounded )
Subset of ( ( ) ( non
empty )
set ) ) st
(
S1 : ( ( non
empty finite ) ( non
empty compact closed finite V147()
V148()
V149()
left_end right_end bounded_below bounded_above real-bounded )
Subset of ( ( ) ( non
empty )
set ) )
= { (dist_min ((Segm (S : ( ( V21() Function-like ) ( V21() Function-like ) set ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( ) ( functional ) Subset of ) ,(Segm (S : ( ( V21() Function-like ) ( V21() Function-like ) set ) ,j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( ) ( functional ) Subset of ) )) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) ) where i, j is ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) : ( 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) <= i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) & i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) < j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) & j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) < len S : ( ( V21() Function-like ) ( V21() Function-like ) set ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) & not i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ,j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) are_adjacent1 ) } &
S2 : ( ( non
empty finite ) ( non
empty compact closed finite V147()
V148()
V149()
left_end right_end bounded_below bounded_above real-bounded )
Subset of ( ( ) ( non
empty )
set ) )
= { (dist_min ((Segm (S : ( ( V21() Function-like ) ( V21() Function-like ) set ) ,(len S : ( ( V21() Function-like ) ( V21() Function-like ) set ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( ) ( functional ) Subset of ) ,(Segm (S : ( ( V21() Function-like ) ( V21() Function-like ) set ) ,k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( ) ( functional ) Subset of ) )) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) ) where k is ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) : ( 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) < k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) & k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) < (len S : ( ( V21() Function-like ) ( V21() Function-like ) set ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) -' 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive V19() V100() V147() V148() V149() V150() V151() V152() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V19() V100() V147() V148() V149() V150() V151() V152() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V147() V148() V149() V150() V151() V152() V153() V181() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V147() V148() V149() V153() V181() non bounded_below non bounded_above interval ) set ) : ( ( ) ( non empty ) set ) ) ) ) } &
it : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real V19()
V100()
V147()
V148()
V149()
V150()
V151()
V152()
bounded_below )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V147()
V148()
V149()
V150()
V151()
V152()
V153()
V181()
left_end bounded_below )
Element of
bool REAL : ( ( ) ( non
empty non
trivial non
finite V147()
V148()
V149()
V153()
V181() non
bounded_below non
bounded_above interval )
set ) : ( ( ) ( non
empty )
set ) ) )
= min (
(min S1 : ( ( non empty finite ) ( non empty compact closed finite V147() V148() V149() left_end right_end bounded_below bounded_above real-bounded ) Subset of ( ( ) ( non empty ) set ) ) ) : ( (
ext-real ) (
V11()
real ext-real )
set ) ,
(min S2 : ( ( non empty finite ) ( non empty compact closed finite V147() V148() V149() left_end right_end bounded_below bounded_above real-bounded ) Subset of ( ( ) ( non empty ) set ) ) ) : ( (
ext-real ) (
V11()
real ext-real )
set ) ) : ( ( ) (
V11()
real ext-real )
set ) );
end;